In this paper, we introduce the definition of a convex real valued function $f$ defined on the set of integers, $\mathbb{Z}$ . We prove that $f$ is convex on $\mathbb{Z}$ if and only if ${{\Delta }^{2}}f\,\ge \,0$ on $\mathbb{Z}$ . As a first application of this new concept, we state and prove discrete Hermite–Hadamard inequality using the basics of discrete calculus (i.e., the calculus on $\mathbb{Z}$ ). Second, we state and prove the discrete fractional Hermite–Hadamard inequality using the basics of discrete fractional calculus. We close the paper by defining the convexity of a real valued function on any time scale.

We study the existence of fixed points for contraction multivalued mappings in modular metric spaces endowed with a graph. The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. This paper can be seen as a generalization of Nadler and Edelstein’s fixed point theorems to modular metric spaces endowed with a graph.

Let $X\left( n \right)$ , for $n\,\in \,\mathbb{N}$ , be the set of all subsets of a metric space $\left( x,\,d \right)$ of cardinality at most $n$ . The set $X\left( n \right)$ equipped with the Hausdorff metric is called a finite subset space. In this paper we are concerned with the existence of Lipschitz retractions $r:\,X\left( n \right)\,\to \,X\left( n\,-\,1 \right)$ for $n\,\ge \,2$ . It is known that such retractions do not exist if $X$ is the one-dimensional sphere. On the other hand, Kovalev has recently established their existence if $X$ is a Hilbert space, and he also posed a question as to whether or not such Lipschitz retractions exist when $X$ is a Hadamard space. In this paper we answer the question in the positive.

The dimensions of the graded quotients of the cohomology of a plane curve complement $U\,=\,{{\mathbb{P}}^{2}}\,\backslash \,C$ with respect to the Hodge filtration are described in terms of simple geometrical invariants. The case of curves with ordinary singularities is discussed in detail. We also give a precise numerical estimate for the difference between the Hodge filtration and the pole order filtration on ${{H}^{2}}\left( U,\,\mathbb{C} \right)$ .

The classification of Euclidean frieze groups into seven conjugacy classes is well known, and many articles on recreational mathematics contain frieze patterns that illustrate these classes. However, it is only possible to draw these patterns because the subgroup of translations that leave the pattern invariant is (by definition) cyclic, and hence discrete. In this paper we classify the conjugacy classes of frieze groups that contain a non-discrete subgroup of translations, and clearly these groups cannot be represented pictorially in any practicalway. In addition, this discussion sheds light onwhy there are only seven conjugacy classes in the classical case.

We use George Bergman's recent normal form for universally adjoining an inner inverse to show that, for general rings, a nilpotent regular element $x$ need not be unit-regular. This contrasts sharply with the situation for nilpotent regular elements in exchange rings (a large class of rings), and for general rings when all powers of the nilpotent element $x$ are regular.

The ${{Q}_{p}}$ spaces of holomorphic functions on the disk, hyperbolic Riemann surfaces or complex unit ball have been studied deeply. Meanwhile, there are a lot of papers devoted to the $Q_{p}^{\#}$ classes of meromorphic functions on the disk or hyperbolic Riemann surfaces. In this paper, we prove the nesting property (inclusion relations) of $Q_{p}^{\#}$ classes on hyperbolic Riemann surfaces. The same property for ${{Q}_{p}}$ spaces was also established systematically and precisely in earlier work by the authors of this paper.

Suppose that $G$ is a finite group and $k$ is a field of characteristic $p\,>\,0$ . A ghost map is a map in the stable category of finitely generated $kG$ -modules which induces the zero map in Tate cohomology in all degrees. In an earlier paper we showed that the thick subcategory generated by the trivial module has no nonzero ghost maps if and only if the Sylow $p$ -subgroup of $G$ is cyclic of order $2$ or $3$ . In this paper we introduce and study variations of ghost maps. In particular, we consider the behavior of ghost maps under restriction and induction functors. We find all groups satisfying a strong form of Freyd’s generating hypothesis and show that ghosts can be detected on a finite range of degrees of Tate cohomology. We also consider maps that mimic ghosts in high degrees.

In this paper, the quaternionic hyperbolic ideal triangle groups are parametrized by a real one-parameter family $\{{{\phi }_{s}}\,:\,s\,\in \,\mathbb{R}\}$ . The indexing parameter s is the tangent of the quaternionic angular invariant of a triple of points in $\partial \mathbf{H}_{\mathbb{H}}^{2}$ forming this ideal triangle. We show that if $s>\sqrt{125/3},$ then ${{\phi }_{S}}$ is not a discrete embedding, and if $s\,\le \,\sqrt{3\text{5}}$ , then ${{\phi }_{S}}$ is a discrete embedding.

We investigate the bi-orderability of two-bridge knot groups and the groups of knots with 12 or fewer crossings by applying recent theorems of Chiswell, Glass and Wilson. Amongst all knots with 12 or fewer crossings (of which there are 2977), previous theorems were only able to determine bi-orderability of 499 of the corresponding knot groups. With our methods we are able to deal with 191 more.

In this note, we study the recurrence and topologically multiple recurrence of a sequence of operators on Banach spaces. In particular, we give a sufficient and necessary condition for a cosine operator function, induced by a sequence of operators on the Lebesgue space of a locally compact group, to be topologically multiply recurrent.

Let $f$ be a holomorphic function of the unit disc $\mathbb{D}$ , preserving the origin. According to Schwarz’s Lemma, $\left| {{f}^{\prime }}(0) \right|\,\le \,1$ , provided that $f(\mathbb{D})\,\subset \,\mathbb{D}$ . We prove that this bound still holds, assuming only that $f(\mathbb{D})$ does not contain any closed rectilinear segment $\left[ 0,\,{{e}^{i\phi }} \right],\,\phi \,\in \,\left[ 0,\,2\pi \right]$ , i.e., does not contain any entire radius of the closed unit disc. Furthermore, we apply this result to the hyperbolic density and give a covering theorem.

Let $R$ be a prime ring of characteristic diòerent from $2$ , let ${{Q}_{r}}$ be its right Martindale quotient ring, and let $C$ be its extended centroid. Suppose that $F$ is a generalized skew derivation of $R,\,L$ a non-central Lie ideal of $R,\,0\,\ne \,a\,\in \,R,\,m\,\ge \,0$ and $n,\,s\,\ge \,1$ fixed integers. If

$$a{{\left( {{u}^{m}}F\left( u \right){{u}^{n}} \right)}^{s}}\,=\,0$$

for all $u\,\in \,L$ , then either $R\,\subseteq \,{{M}_{2}}\left( C \right)$ , the ring of $2\,\times \,2$ matrices over $C$ , or $m\,=\,0$ and there exists $b\,\in \,{{Q}_{r}}$ such that $F\left( x \right)\,=\,bx$ , for any $x\,\in \,R$ , with $ab\,=\,0$ .

For $T$ a compact torus and $E_{T}^{*}$ a generalized $T$ -equivariant cohomology theory, we provide a systematic framework for computing $E_{T}^{*}$ in the context of equivariantly stratified smooth complex projective varieties. This allows us to explicitly compute $H_{T}^{*}\left( X \right)$ as an $H_{T}^{*}\left( pt \right)$ -module when $X$ is a direct limit of smooth complex projective ${{T}_{\mathbb{C}}}$ -varieties. We perform this computation on the affine Grassmannian of a complex semisimple group.

The thickness of a graph $G$ is the minimum number of planar subgraphs whose union is $G$ . A $t$ -minimal graph is a graph of thickness $t$ that contains no proper subgraph of thickness $t$ . In this paper, upper and lower bounds are obtained for the thickness, $t\left( G\,\square \,H \right)$ , of the Cartesian product of two graphs $G$ and $H$ , in terms of the thickness $t\left( G \right)$ and $t\left( H \right)$ . Furthermore, the thickness of the Cartesian product of two planar graphs and of a $t$ -minimal graph and a planar graph are determined. By using a new planar decomposition of the complete bipartite graph ${{K}_{4k,\,4k}}$ , the thickness of the Cartesian product of two complete bipartite graphs ${{K}_{n,n}}$ and ${{K}_{n,n}}$ is also given for $n\,\ne \,4k\,+\,1$ .

We prove that the existence spectrum of Mendelsohn triple systems whose associated quasigroups satisfy distributivity corresponds to the Loeschian numbers, and provide some enumeration results. We do this by considering a description of the quasigroups in terms of commutative Moufang loops. In addition we provide constructions of Mendelsohn quasigroups that fail distributivity for asmany combinations of elements as possible. These systems are analogues of Hall triple systems and anti-mitre Steiner triple systems respectively.

Let $R\,=\,{{\oplus }_{n\ge 0}}{{R}_{n}}$ be a graded Noetherian ring with local base ring $\left( {{R}_{0}},{{\text{m}}_{0}} \right)$ and let ${{R}_{+}}\,=\,{{\oplus }_{n>0}}{{R}_{n}}$ . Let $M$ and $N$ be finitely generated graded $R$ -modules and let $\mathfrak{a}\,=\,{{\mathfrak{a}}_{0}}\,+\,{{R}_{+}}$ an ideal of $R$ . We show that $H_{\mathfrak{b}0}^{j}\,\left( H_{\mathfrak{a}}^{i}\left( M,\,N \right) \right)$ and ${H_{\mathfrak{a}}^{i}\left( M,\,N \right)}/{{{\mathfrak{b}}_{0}}H_{\mathfrak{a}}^{i}\left( M,\,N \right)}\;$ are Artinian for some $i\text{ s}$ and $j\,\text{s}$ with a specified property, where ${{\mathfrak{b}}_{o}}$ is an ideal of ${{R}_{0}}$ such that ${{\mathfrak{a}}_{0}}\,+\,{{\mathfrak{b}}_{0}}$ is an ${{\mathfrak{m}}_{0}}$ -primary ideal.

There are several kinds of classification problems for real hypersurfaces in complex two-plane Grassmannians ${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$ . Among them, Suh classified Hopf hypersurfaces in ${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$ with Reeb parallel Ricci tensor in Levi–Civita connection. In this paper, we introduce the notion of generalized Tanaka–Webster $\left( \text{GTW} \right)$ Reeb parallel Ricci tensor for Hopf hypersurfaces in ${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$ . Next, we give a complete classification of Hopf hypersurfaces in ${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$ with $\text{GTW}$ Reeb parallel Ricci tensor.

In this note we present a simple alternative proof for the Bernstein problem in the three dimensional Heisenberg group $\text{Ni}{{\text{l}}_{3}}$ by using the loop group technique. We clarify the geometric meaning of the two-parameter ambiguity of entire minimal graphs with prescribed Abresch-Rosenberg differential.

We show that the discrete Hilbert transform and the discrete Kak–Hilbert transform are infinitesimal generators of one-parameter groups of operators in ${{\ell }^{2}}$ .